2.2 Patterns & Inductive Reasoning Geometry 2.2 Patterns & Inductive Reasoning Essential Question: How can you prove a conjecture false?

Geometry 2.2 Patterns and Inductive Reasoning Topic/Objectives: Find and describe patterns. Use inductive reasoning to make conjectures. To understand “What is a conjecture” Know what a counterexample is. April 6, 2019 Geometry 2.2 Patterns and Inductive Reasoning

Geometry 2.2 Patterns and Inductive Reasoning Find a Pattern What’s the next number? 1) 17, 15, 12, 8, … 3 2) 1, -2, 4, -8, … 16 3) 48, 12, 3, … 3/4 Now check your answers. April 6, 2019 Geometry 2.2 Patterns and Inductive Reasoning

Geometry 2.2 Patterns and Inductive Reasoning You have just done Inductive Reasoning April 6, 2019 Geometry 2.2 Patterns and Inductive Reasoning

Geometry 2.2 Patterns and Inductive Reasoning Look for a pattern Make a conjecture Conjecture: An unproven statement based on observations. Verify the conjecture Inductive Reasoning is making conjectures based on patterns April 6, 2019 Geometry 2.2 Patterns and Inductive Reasoning

Geometry 2.2 Patterns and Inductive Reasoning Making a Conjecture The sum of two odd numbers is _____. Think of some odd numbers: 3 5 9 11 21 13 7 Now think of the sums of some of them: even See any Pattern? 5 + 21 = 26 3 + 5 = 8 even 7 + 9 = 16 even even 11 + 13 = 24 April 6, 2019 Geometry 2.2 Patterns and Inductive Reasoning

Geometry 2.2 Patterns and Inductive Reasoning Our Conjecture: The sum of any two odd numbers is EVEN. April 6, 2019 Geometry 2.2 Patterns and Inductive Reasoning

Your turn. Study the examples and make a conjecture. 1 = 1 1 + 3 = 4 1 + 3 + 5 = 9 1 + 3 + 5 + 7 = 16 The sum of the first n odd integers is____. 12 22 32 42 n2 April 6, 2019 Geometry 2.2 Patterns and Inductive Reasoning

Geometry 2.2 Patterns and Inductive Reasoning To prove a conjecture is TRUE, you must be able to prove it for all cases. Proof April 6, 2019 Geometry 2.2 Patterns and Inductive Reasoning

To prove a conjecture is false, you need only one counterexample. April 6, 2019 Geometry 2.2 Patterns and Inductive Reasoning

Geometry 2.2 Patterns and Inductive Reasoning Counterexample An example that proves a statement to be false. April 6, 2019 Geometry 2.2 Patterns and Inductive Reasoning

Geometry 2.2 Patterns and Inductive Reasoning Example Conjecture: The sum of any two integers is positive. BUT… 1 + 5 = 6 -4 + 11 = 7 0 + 1 = 1 -12 + 100 = 88 -7 + 4 = -3 which is not positive. So, we found a counterexample, and the conjecture is false. April 6, 2019 Geometry 2.2 Patterns and Inductive Reasoning

Your turn. Find a counterexample to the statement. For all real numbers, n2 > n. Counterexamples: 12 is not > 1 02 is not > 0 0.52 is not > 0.5 9 > 3 25 > 5 100 > 10 April 6, 2019 Geometry 2.2 Patterns and Inductive Reasoning

Geometry 2.2 Patterns and Inductive Reasoning Unproven Conjectures Conjectures that have not been proven true, but no counterexample can be found. April 6, 2019 Geometry 2.2 Patterns and Inductive Reasoning

Goldbach’s Conjecture Every even number greater than 2 can be written as the sum of two prime numbers. Prime number: only divisible by 1 and itself. 2, 3, 5, 7, 11, 13, 17, 19… April 6, 2019 Geometry 2.2 Patterns and Inductive Reasoning

Goldbach’s Conjecture Every even number greater than 2 can be written as the sum of two prime numbers. 6 = 3 + 3 8 = 3 + 5 18 = 20 = 3 + 17 24 = 5 + 19 11 + 13 5 + 13 April 6, 2019 Geometry 2.2 Patterns and Inductive Reasoning

Geometry 2.2 Patterns and Inductive Reasoning Christian Goldbach (1690 – 1764) made his conjecture almost 300 years ago, and it still hasn’t been proven, despite efforts by some of the greatest mathematicians of all time. Many people today suspect that it is true, but that it cannot, and never will, be proven true. April 6, 2019 Geometry 2.2 Patterns and Inductive Reasoning

Geometry 2.2 Patterns and Inductive Reasoning Important! Just because something is true for several specific cases does not prove that it is true in general. April 6, 2019 Geometry 2.2 Patterns and Inductive Reasoning

Geometry 2.2 Patterns and Inductive Reasoning What have you learned? Take a moment, and quietly write down one thing you have learned today that you didn’t know before. April 6, 2019 Geometry 2.2 Patterns and Inductive Reasoning